Been riding with our dog via trailer, primarily on bike paths for about 10 years, without incident...unless you count the two times I took corners too fast and rolled the trailer. Both dog and trailer were fine. Kids in a trailer the 15 years before that. Nobody died or got concussed. Nothing and nobody in a trailer had helmets on.
This trailer is specifically for dogs. We have a beach house down south where the kids trailer is used for the dog, sans seats. That trailer has serious history and has been in use since 1996.